Recently, Large Language Models (LLMs) have made substantial progress in code generation, but they still frequently generate code containing logic errors or syntax bugs. While research has focused on improving performance through fine-tuning and data collection, less attention has been given to analyzing error patterns and employing prompt-engineering to address these issues. Existing benchmarks primarily assess LLMs on easy to intermediate-level coding tasks, often neglecting more complex challenges involving advanced algorithms and data structures. Additionally, data contamination in these benchmarks limits their ability to accurately measure the capability of LLMs in code generation. In this paper, we present the new USACO Benchmark, derived from the USA Computing Olympiad (US-ACO) competition, to evaluate 11 closed and open-source LLMs. Through a detailed analysis, we identify common code generation errors across the models and propose Hint-Driven Prompts to address logic errors, alongside the Syntax Mitigation Prompt to reduce syntax bugs. Our results demonstrate that the Hint-Driven Prompt boosts pass rates for DBRX 132B, Deepseek-Coder 33B, Codegemma 7B, Codellama 7B, Llama 3, and GPT-4o by 6.6x, 4.7x, 3x, 2.5x, 2.1x, and 25%, respectively. Additionally, the Syntax Mitigation Prompt significantly reduces syntax errors, with reductions of 71.32% for Codegemma 7B, 25.56% for Deepseek-Coder 33B, 23.39% for Llama 3, and 11.19% for Codellama 70B.